**Date & Time:** Monday, June 27, 2011,
16:00 - 17:00

**Venue:**Ramanujan Hall, Department of Mathematics

**Title:** Metric Projections in Spaces of Continuous Functions

**Speaker:** Professor A.L. Brown of the University College London (U.K.)

**Abstract: **
Let T be a topological space (a compact subspace of Rm , say) and let C(T )
be the space of real continuous functions on T , equipped with the uniform
norm: f = maxt∈T |f (t)| for all f ∈ C(T ). Let G be a finite dimensional
linear subspace of C(T ). If f ∈ C(T ) then
d(f, G) = inf{ f − g : g ∈ G}
is the distance of f from G, and
PG (f ) = {g ∈ G : f − g = d(f, G)}
is the set of best approximations to f from G. Then
PG : C(T ) → P(G)
is the set-valued metric projection of C(T ) onto G. In the 1850s P. L. Chebyshev
considered T = [a, b] and G the space of polynomials of degree ≤ n − 1. Our
concern is with possible properties of PG . The historical development, beginning
with Chebyshev, Haar (1918) and Mairhuber (1956), and the present state of
knowledge will be outlined. New results will demonstrate that the story is still
incomplete.